Related papers: The attractor structure of logarithmic iterations …
We prove that under certain combinatorial conditions, the realization spaces of line arrangements on the complex projective plane are connected. We also give several examples of arrangements with eight, nine and ten lines which have…
In these lectures we discuss some of the mathematical structures that appear when computing multi-loop Feynman integrals. We focus on a specific class of special functions, the so-called multiple polylogarithms, and discuss introduce their…
We survey results in algebraic complexity theory, focusing on matrix multiplication. Our goals are (i.) to show how open questions in algebraic complexity theory are naturally posed as questions in geometry and representation theory, (ii.)…
In this paper, plane polynomial systems having a singular point attracting all orbits in positive time are classified up to topological equivalence. This is done by assigning a combinatorial invariant to the system (a so-called "feasible…
The well known butterfly effect got its nomenclature from its two wings geometrical structure in phase space. There are chaotic dynamics from simple one-wing to multiple-wings complex structures in phase space. In this communication we…
In this paper, we revisit the chaotic number of iterations needed by Newton's method to converge to a root. Here, we consider a simple modified Newton method depending on a parameter. It is demonstrated using polynomiography that even in…
We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study…
The Law of the Iterated Logarithm for some Markov operators, which converge exponentially to the invariant measure, is established. The operators correspond to iterated function systems which, for example, may be used to generalize the cell…
We classify the measure theoretic attractors of general C^3 unimodal maps with quadratic critical points. The main ingredient is the decay of geometry.
Using vertex algebra techniques, we determine a set of generators for the cohomology ring of the Hilbert schemes of points on an arbitrary smooth projective surface over the field of complex numbers.
We study some natural operators acting on configurations of points and lines in the plane and remark that many interesting configurations are fixed points for these operators. We review ancient and recent results on line or point…
The structure of the Lorenz-84 attractor is investigated in this study. Its dynamics belonging to weakly dissipative chaos, classical approaches cannot be used to analyze its structure. The color tracer mapping is introduced for this…
This paper deals with analytic families of holomorphic iterated function systems. Using real analyticity of the pressure function (which we prove), we establish a classification theorem for analytic families of holomorphic iterated function…
Our aim is to reprove the basic results of the theory of branches of plane algebraic curves over algebraically closed fields of arbitrary characteristic. We do not use the Hamburger-Noether expansions. Our basic tool is the logarithmic…
We discuss the structure of integral etale motivic cohomology groups of smooth and projective schemes over algebraically closed fields, finite fields, local fields, and arithmetic schemes.
A study of rational maps of the real or complex projective plane of degree two or more, concentrating on those which map an elliptic curve onto itself, necessarily by an expanding map. We describe relatively simple examples with a rich…
We introduce and analyze a new geometric structure on topological surfaces generalizing the complex structure. To define this so called higher complex structure we use the punctual Hilbert scheme of the plane. The moduli space of higher…
Conditions are given which imply that certain non-autonomous analytic iterated function systems (NIFS's) in the complex plane C have uniformly perfect attractor sets. Examples are given to illustrate the main theorem, as well as to indicate…
We describe some results on moduli space of logarithmic connections equipped with framings on a $n$-pointed compact Riemann surface.
We point out how Banach Fixed Point Theorem, and the Picard successive approximation methods induced by it, allows us to treat some mathematical methods in Combinatorics. In particular we get, by this way, a proof and an iterative algorithm…